The last step is either false or misleading nobody would write that. It means in one case ++x and in the other case --x which is both the same as x and not what you wrote on the line above. Plus-minus signs are used if one wants to say that there are two equations that apply in different cases, and that are only distinguished by different plus or minus signs. It's a way to express your thoughts and not a rigorous 100% correct way to write mathematics. If one wants to be clear one has to write cases like

Plus/minus signs are not aways used in front of radicals. The n-th root is a function which means it can produce only one output. In the case where n=even number the square root is positive number which rised to the power of n gives the original number. However negative numbers risen to even power give the same answer as the same positive numbers. When you solve equations you usualy need both cases.

This last step is incorrect. [itex]\pm\sqrt{x^2}[/itex] is the same as [itex]\pm |x|[/itex] but |x| is NOT the same as "[itex]\pm x[/itex]" so [itex]\pm |x|[/itex] is NOT the same as "[itex]\pm\pm x[/itex]".